ne_bot_of_isCartanSubalgebra — Mathlib

English

As before, a nontrivial Cartan subalgebra cannot be the bottom subalgebra.

Русский

Как и прежде, неtrivial Картанова подалгебра не может быть нулевой подалгеброй.

LaTeX

$$$H \\neq \\bot$ under [Nontrivial L] and [H.IsCartanSubalgebra]$$

Lean4

theorem ne_bot_of_isCartanSubalgebra [Nontrivial L] (H : LieSubalgebra R L) [H.IsCartanSubalgebra] : H ≠ ⊥ :=
  by
  intro e
  obtain ⟨x, hx⟩ := exists_ne (0 : L)
  have : x ∈ H.normalizer := by simp [LieSubalgebra.mem_normalizer_iff, e]
  exact hx (by rwa [LieSubalgebra.IsCartanSubalgebra.self_normalizing, e] at this)

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